What about integration of scalar/vector fields with respect to vectors. I am interested in how you solve these:

Where f is a scalar field and

Where g is a vector field.

Any input is appreciated.

Dec 21st 2008, 08:40 AM

Mush

Quote:

Originally Posted by fobos3

Let r=xi+yj+zk be a displacement vector. We have:

Where f is a scalar field

What if f is a vector field. Does that mean that:

What about integration of scalar/vector fields with respect to vectors. I am interested in how you solve these:

Where f is a scalar field and

Where g is a vector field.

Any input is appreciated.

is the gradient of scalar field f. A vector does not have a gradient in this sense. A vector can have curl and divergence, but not gradient. The gradient of a scalar field is a vector field.

Nabla, is defined as :

If you have a vector , then you can have only two multiplicative operations with this vector and the nabla vector. Those are the dot product ( )and the cross product ( ). And these represent divergence and curl respectively. It does not make sense to have scalar multiplcation between two vectors, which is what you are proposing with .

You may have scalar multiplication between two scalars, and scalar multiplcation between a scalar and a vector (a lá, , but you may not have scalar multiplication between two vectors.

Dec 21st 2008, 10:55 AM

fobos3

Quote:

Originally Posted by Mush

is the gradient of scalar field f. A vector does not have a gradient in this sense. A vector can have curl and divergence, but not gradient. The gradient of a scalar field is a vector field.

Nabla, is defined as :

If you have a vector , then you can have only two multiplicative operations with this vector and the nabla vector. Those are the dot product ( )and the cross product ( ). And these represent divergence and curl respectively. It does not make sense to have scalar multiplcation between two vectors, which is what you are proposing with .

You may have scalar multiplication between two scalars, and scalar multiplcation between a scalar and a vector (a lá, , but you may not have scalar multiplication between two vectors.